lzl='. Integrated Short Term Navigation of a Towed Underwater Body* G. Damy' M. Joannides2 F. LeGland3 M. Prkvosto' R. Rakotozafy2

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1 Integrated Short Term Navigation of a Towed Underwater Body* G. Damy' M. Joannides2 F. LeGland3 M. Prkvosto' R. Rakotozafy2 IFREMER Brest, DITI/GO/SOM, BP 70, PLOUZANE, France INRIA Sophia Antipolis, BP 93, SOPHIA-ANTIPOLIS Ckdex, France IRISA / INRIA Rennes, Campus de Beaulieu, RENNES CCdex, France I. POSITION OF THE PROBLEM The following ocean engineering problem is considered. An underwater body, to be called hereafter the fish, is towed by a surface ship at the end of a few hundred meters long cable, and uses its sidescan sonars to make observations of the bottom of the sea, a few thousand meters below. To enhance the antenna resolution, a virtual antenna is built up by adding up appropriately the echoes received by the moving physical antenna. To make this SAS (synthetic aperture sonar) experiment efficiently, it is needed to know with an accuracy of a few centimeters, the trajectory of the fish relative to its otherwise unknown initial position, during a few minutes experiment, so that motion compensation can be performed during the synthetic aperture processing. For this purpose, acceleration measurements provided by an INS (inertial navigation system) located on board of the fish, can be integrated. INS measurements are known to track accurately the high frequency components in the mobile trajectory. However, even on small time intervals, integration of INS measurements can result in unacceptable drifting in position estimates, i.e. poor estimation of low frequency components of the mobile trajectory. To prevent this drifting in position estimates, it is common to use additional measurements which could provide good estimates of the low frequency components of the mobile trajectoay. For example, the following additional sensors could be used : (i) a quartz pressure sensor, providing submersion measurements, (ii) an electromagnetic velocity probe, (iii) a magnetic heading sensor, etc. Combining the two complementary type of measurements would usually improve very much the trajectory estimates. This can be performed b!y means of a Kalman filter, see e.g. Maybeck [l, Chap. 61. The purpose of this paper is to investigate the possibility of using position estimates of the surface ship, provided by differential GPS (global po- *This work was partially supported by IFREMER, under contract no sitioning system). An additional difficulty is that the satellites involved in GPS cannot provide any position estimates of an underwater body such as a towed fish, but only position estimates of the surface ship. To overcome this difficulty, the idea is to introduce a numerical model for the towing cable, in order to transform GPS position measurements of the surface ship into position and velocity estimates of the towed fish, so as to be able to combine these estimates with INS measurements. The modelization of the cablefish system is briefly discussed in Section 11.. Numerical results are reported in Section III., which illustrate the interest; and feasibility of the proposed hybridation procedure. I1 MODELIZATION OF THE CABLE-FISH SYSTEM For the sake of simplicity, it is assumed here that the surface ship is heading in a constant fixed direction, during the considered period of time. Moreover, it is assumed that the cable initial profile, and its subsequent deformations are all contained in the vertical plane defined by the surface ship heading direction. The notations in use are introduced in Tables I, 11, and 111. By convention, the curvilinear coordinate s = 0 corresponds to the fish, and related quantities are indexed with the letter F, e.g. VZF will denote the mass of the fish in the water. Similarly, the curvilinear coordinate s = L corresponds to the towing point on the surface ship, and related quantities are indexed with the letter S, e.g. FS and Cs will denote the position and velocity of the surface ship. Since the cable is short enough (a few hundred meters), it is reasonable to assume that it is znextensible, which is expressed by the constraint ar lzl='. Also, it is assumed that the cable is infinitely Bexible, which results in the cable tension to be aligned IEEE

2 ~~~ with the tangent vector, i.e. F=T7'. The last modelling assumption is that there is no tangential component of the hydrodynamical drag force per unit length, i.e. - D=Dn'. The following expression is used for the normal component D of the hydrodynamical drag force per unit length at the current cable point D = -12 PO cd d lvll VI (1) where ~1 = 5- n' is the normal component of the velocity at the current cable point, and where the constants are defined in Table I. Let c' denote the sum of all external forces per unit length which are applied at the current cable point, i.e. c'=p [l- -1UP0 ;+E P The second law of motion results in the following?volution equation L : d : U : s : r ' : 3 r ' : 7 : n ' : P : F : E : length diameter section (U = +mp) curvilinear coordinate position velocity acceleration tangent vector to the cable normal vector to the cable mass per unit length cable tension hydrodynamical drag force per unit length empirical drag coefficient Table I Notations for the cable. : gravitational acceleration po : water mass per unit volume Table I1 Other notations. In addition, the position and velocity of the surface ship are given (or measured) at any time, which provides the boundary conditions at s = L. Similarly, let 2~ denote the sum of all external forces which are applied to the fish, i.e. where & is the hydrodynamical drag force. The following expression is used for the hydrodynamical - drag force d~ applied to the fish ZF = -4 PO Cd,F AF IGFl CF, (2) where both the drag coefficient Cd,F and the effective lateral cross-section area AF are diagonal matrices, and where the constants are defined in Table 111, see Ivers and Mudie [3]. The second law of motion results in the following evolution equation - + TF + CF = [mf + ma,f]?f, where the added mass ma,f is a diagonal matrix. This provides the desired boundary condition at s = 0. EF Cd,F : position : velocity : acceleration : mass : added mass : mass per unit volume : effective lateral : cross-section area : hydrodynamical drag force : empirical drag coefficient Table 111 Notations for the fish. L : m d : 12.3 mm p : 0.6 kg/m c d : 1.8 mf : 2000 kg p~ : 1000 kg/m3 Cd,F : 0.5 Table IV Main numerical values for the simulation

3 To summarize, the unknown variables in this qroblem are position F, velocity v', and cable tension T at each point of the cable. The model equations are a -(TF)+p[l--]g'+Dii as P = py for 0 5 s 5 L. The boundary conditions at s = 0 and s = L are respectively arid where R' and f are given (or estimated) position and velocity of the surface ship. Finally, the expressions for the drag coefficients D and & are given in (1) and (2) respectively. A numerical approximation scheme for the model equations has been presented in Joannides and LeGland [2] NUMERICAL EXPERIMENT The following simulation experiment is reported here, which illustrates the interest and feasibility of the proposed hybridation procedure. The main numerical values for the cablefish model are given in Table IV. In particular, it should be noticed that the fish is neutrally buoyant, since it has the same mass per unit volume as water. The intuitive effect which is expected from this design, is to make the trajectory of the towed fish as independent as possible from the high frequency vertical oscillations of the surface ship trajectory. This effect can be verified from the numerical simulation reported here. A.. Generation of reference trajectories Let v* = (1.5, 0.0) denote some nominal constant velocity (units in m/s). The horizontal velocity of the surface ship has a low frequency sinusoidal component, with period 60 s and amplitude 0.1 m/s, around the nominal constant velocity 1.5 m/s. The vertical velocity of the surface ship has a high frequency sinusoidal component, with period 5 s and amplitude m/s, around the nominal constant velocity O m/s. At time to = 0.0 s, the position and velocity of the surface ship are set to ro = (-0.955, 0.5) (units in m), and vo = U* = (1.5, 0.0) (units in m/s), respectively. The reference trajectory of the fish is obtained by integrating the cablefish numerical model : The input (boundary condition at the upper end of the cable) is set to the reference trajectory of the surface ship, obtained by the procedure described ab ovls. At time to = 0.0 s, the cable profile is set to the permanent profile associated with the nominal constant velocity U* = (1.5, 0.0) (units in m/s), with the upper end of the cable (corresponding to the towing point on the surface ship) translated to the position ro = (-0.955, 0.5) (units in m) of the surface ship at time to. B. Generation of INS measurements From the reference trajectory of the fish, obtained by the procedure described in Subsection A. above, noisy measurements rlns of the acceleration Yk of the fish at time tins are generated in additive white noise wlns, with standard deviation 0.01 m/s2 (both horizontal and vertical), and period AINs = 20 ms = 0.02 s. AINS tins - E+1 - tk INS C. Integration of INS measurements The noisy acceleration measurements of the fish, generated by the procedure described in Subsection B. above are integrated to produce INS estimates of the velocity of the fish, according to the integration scheme VINS INS + AINS I k+l - vk rlns Ai& time tins, the initial condition of the integration scheme is set to the nominal constant velocity VINS - v * - (1.5, 0.0) (units in m/s). In general, uins is different from the true velocity of the fish at time tinns This introduces an initial bias in velocity estimates of the fish, which results in a linear drift in position estimates of the fish. This is illustrated by Figures 4 and 6, and by Figures 5 and 7 respectively

4 D. Generation of GPS measurements From the reference trajectory of the surface ship, obtained by the procedure described in Subsec.. tion A. above, noisy measurements rfps of the position rk of the surface ship at time tfps, are generated in additional white noise w ~ps, with standard deviation 0.2 m (horizontal) and 0.5 m (vertical), and period AGPS = 0.6 s.,.gps k -rk+wfps AGPS - tgps - tgps - k+1 k E. Kalman filtering of GPS measurements Smooth inputs are needed for the cable-fish nu-. merical model, with sampling period 0.01 s. For this purpose, a Kalman filter is used, which combines the model described in Subsection A above for the velocity of the surface ship (sinusoidal component with known frequency around nominal velocity), and the noisy position measurements of the surface ship, generated by the procedure de scribed in Subsection D. above. At time to = 0.0 s, the initial conditions OS the Kalman filter estimates are set to the nominal values, i.e. PO = (0.0, 0.0) (units in m), and h WO = U* = (1.5, 0.0) (units in m/s). F. Generation of GPS-based velocity estimates of the fish The GPS-based trajectory of the fish is obtained by integrating the cablefish numerical model : The input (boundary condition at the upper end of the cable) is set to the Kalman filter position and velocity estimates of the surface ship, obtained by the procedure described in Subsection E. above. As can be seen from Figures 1,2 and 3, these estimates become very accurate after a short transient period. Therefore, the initial time for subsequent computations (integration of the cable-fish numerical model, integration and hybridation of INS acceleration measurements) is set to t'ons = 60.0 s. At time ttns = 60.0 s, the cable profile is set to the permanent profile associated with the constant nominal velocity U* = (1.5, 0.0) (units in m/s), with the upper end of the cable (corresponding to the towing point on the surface ship) translated to the Kalman filter position estimate of the surface ship at time tlns Results are presented in Figures 8 and 9 Figure 1. Horizontal position of surface ship : reference trajectory and GPS measurements j Figure 2. Horizontal position of surface ship : Figure 3. Horizontal velocity of surface ship : I im

5 Figure 7. Vertical position of towed fish : reference trajectory and INS-based estimate..... m m im im iui 180 im 0 OB 1 im im 180 Figure 5. Horizontal position of towed fish : reference trajectory and INS-based estimate Figure 8. Vertical velocity of towed fish : n "C I. I I I I 0 OB m m rm 120 I im Figure 6. Vertical velocity of towed fish : reference trajectory and INS-based estimate.rma. i.-. L. j. j -104 L a m im im 140 1M Figure 9. Vertical position of towed fish : a

6 G. Hybridation To correct the initial bias in velocity estimates of the fish, the INS acceleration measurements arfo combined with the GPS-based velocity estimates obtained by the procedure described in Subsection F. above, according to the following hybridation procedure I Gk+1 vins = ijk + AINS.,,ZNS k+l INS = 6k vk+l + (1 - e k) which is an adaptive modification of the original integration procedure described in Subsection C. above. The optimal weighting coefficients 8k could be obtained by performing a careful sensitivity analysis of the cable-fish numerical model. The result presented in Figure 10 corresponds rather to constant weighting coefficients 8k E 8, for a constani; 5, value 8 which has been selected by try and guess. IV. CONCLUSION Although the numerical results presented here are very promising, the following two comment:; should be made. mathematical model for the surface ship evolution. This will never be the case in real sit uation. 0 The selection of a constant weighting coefficient obtained by try and guess for the hybridation of INS acceleration measurements and GPS-based velocity estimates is not acceptable. The determination of optimal weighting coefficients is under investigation, and will be presented elsewhere. REFERENCES [l] P. MAYBECK, Stochastzc Models, Estzmatzon, and Control. Volume 1, vol of Matheniatzcs zn Sczence and Engzneerzng. New York: Academic Press, [a] M. JOANNIDES and F LE GLAND, POSItion estimation of a towed underwater body, in Proceedangs of the 32nd IEEE Conference on Decaszon and Control, San Antonzo 1993, pp , IEEE, Dec [3] W. IVERS and J. MUDIE, Towing a long cable at slow speeds : A three-dimensional dynamic model, Martne Technology Soczely Journal, vol. 7, no. 3, p , i I i 1 1 m Bo rm im 140 im 180 Figure 10. Vertical position of towed fish : reference trajectory and hybrid estimate